On the persistence and unique continuation properties for an integrable two-component Dullin–Gottwald–Holm system

Considered herein is the persistence and unique continuation properties of the solution of an integrable two-component Dullin–Gottwald–Holm (DGH2) system, which was derived from the Euler equation with constant vorticity in shallow water waves moving over a linear shear flow. It is proved that the solutions of the DGH2 system enjoy the same exponential decay property as the initial data, as a consequence, if the solutions and their spacial derivatives decay exponentially initially and at a later time, then they must identically equal to zero.